space curve

Kinematic definition.

A parameterized space curve is a parameterized curve taking values in 3-dimensional Euclidean space. It may be interpreted as the trajectory of a particle moving through space. Analytically, a smooth space curve is represented by a sufficiently differentiable mapping $\gamma :I\to {ℝ}^3,$ of an interval $I\subset ℝ$ into 3-dimensional Euclidean space ${ℝ}^3$. Equivalently, a parameterized space curve can be considered a 3-vector of functions:

$$\gamma (t)=\left(\begin{array}{c}\hfill x(t)\hfill \\ \hfill y(t)\hfill \\ \hfill z(t)\hfill \end{array}\right),t\in I.$$

Regularity hypotheses.

To preclude the possibility of kinks and corners, it is necessary to add the hypothesis that the mapping be regular (http://planetmath.org/Curve), that is to say that the derivative ${\gamma }^{\prime }(t)$ never vanishes. Also, we say that $\gamma (t)$ is a point of inflection if the first and second derivatives ${\gamma }^{\prime }(t),{\gamma }^{\prime \prime }(t)$ are linearly dependent. Space curves with points of inflection are beyond the scope of this entry. Henceforth we make the assumption that $\gamma (t)$ is both and lacks points of inflection.

Geometric definition.

A space curve, per se, needs to be conceived of as a subset of ${ℝ}^3$ rather than a mapping. Formally, we could define a space curve to be the image of some parameterization $\gamma :I\to {ℝ}^3$. A more useful concept, however, is the notion of an oriented space curve, a space curve with a specified direction of motion. Formally, an oriented space curve is an equivalence class of parameterized space curves; with ${\gamma }_1:I_1\to {ℝ}^3$ and ${\gamma }_2:I_2\to {ℝ}^3$ being judged equivalent if there exists a smooth, monotonically increasing reparameterization function $\sigma :I_1\to I_2$ such that

$${\gamma }_1(t)={\gamma }_2(\sigma (t)),t\in I_1.$$

Arclength parameterization.

We say that $\gamma :I\to {ℝ}^3$ is an arclength parameterization of an oriented space curve if

$$\parallel {\gamma }^{\prime }(t)\parallel =1,t\in I.$$

With this hypothesis the length of the space curve between points $\gamma (t_2)$ and $\gamma (t_1)$ is just $|t_2-t_1|$. In other words, the parameter in such a parameterization measures the relative distance along the curve.

Starting with an arbitrary parameterization $\gamma :I\to {ℝ}^3$, one can obtain an arclength parameterization by fixing a $t_0\in I$, setting

$$\sigma (t)={\int }_{t_0}^t\parallel {\gamma }^{\prime }(x)\parallel 𝑑x,$$

and using the inverse function ${\sigma }^{-1}$ to reparameterize the curve. In other words,

$$\widehat{\gamma }(t)=\gamma ({\sigma }^{-1}(t))$$

is an arclength parameterization. Thus, every space curve possesses an arclength parameterization, unique up to a choice of additive constant in the arclength parameter.

Title	space curve
Canonical name	SpaceCurve
Date of creation	2013-03-22 12:15:03
Last modified on	2013-03-22 12:15:03
Owner	Mathprof (13753)
Last modified by	Mathprof (13753)
Numerical id	15
Author	Mathprof (13753)
Entry type	Definition
Classification	msc 53A04
Synonym	oriented space curve
Synonym	parameterized space curve
Related topic	Torsion
Related topic	CurvatureOfACurve
Related topic	MovingFrame
Related topic	SerretFrenetFormulas
Related topic	Helix
Defines	point of inflection
Defines	arclength parameterization
Defines	reparameterization